A dimension reduction approach for conditional Kaplan–Meier estimators

Many quantities of interest in survival analysis are smooth, closed-form functionals of the law of the observations. For instance, the conditional law of a lifetime of interest under random right censoring, and the conditional probability of being cured. In such cases, one can easily derive nonparametric estimators for the quantities of interest by plugging-into the functional the nonparametric estimators of the law of the observations. However, with multivariate covariates, the nonparametric estimation suffers from the curse of dimensionality. Here, a new dimension reduction approach for survival analysis is proposed and investigated in the right-censored lifetime case. First, we consider a single-index hypothesis on the conditional law of the observations and propose a n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}-$$\end{document}asymptotically normal semiparametric estimator. Next, we apply the smooth functionals to this estimator. This results in semiparametric estimators of the quantities of interest that avoid the curse of dimensionality. Confidence regions for the index and the functional of interest are built by bootstrap. The new methodology allows to test the dimension reduction assumption, can be extended to other dimension reduction methods and can be applied to closed-form functionals of more general censoring mechanisms.